Tarek Garna

Decomposition of an ARX model on Laguerre orthonormal bases.

In this paper, we propose a new reduced complexity model by expanding a discrete-time ARX model on Laguerre orthonormal bases. To ensure an efficient complexity reduction, the coefficients associated to the input and the output of the ARX model are expanded on independent Laguerre bases, to develop a new black-box linear ARX-Laguerre model with filters on model input and output. The parametric complexity reduction with respect to the classical ARX model is proved theoretically. The structure and parameter identification of the ARX-Laguerre model is achieved by a new proposed approach which consists in solving an optimization problem built from the ARX model without using system input/output observations. The performances of the resulting ARX-Laguerre model and the proposed identification approach are illustrated by numerical simulations and validated on benchmark manufactured by Feedback known as Process Trainer PT326. A possible extension of the proposed model to a multivariable process is formulated.

Online identification of the ARX model expansion on Laguerre orthonormal bases with filters on model input and output

This article proposes a new representation of the ARX models on independent and orthonormal Laguerre bases by filtering the process input and output using Laguerre orthonormal functions. The resulting model, entitled ARX–Laguerre model, ensures the parameter number reduction with a recursive and easy representation. However, this reduction is still subject to an optimal choice of the Laguerre poles defining both Laguerre bases. Therefore, we propose an analytical solution to optimise the Laguerre poles which depend on Fourier coefficients defining the ARX–Laguerre model, and that are identified using the regularised square error. The identification procedures of the Laguerre poles and Fourier coefficients are combined and carried out on a sliding window to provide an online identification algorithm of the ARX–Laguerre model. The proposed algorithm is tested on numerical simulation and validated on a benchmark system manufactured by Feedback known as Process Trainer PT326.

Non-linear predictive controller for uncertain process modelled by GOBF-Volterra models

This paper proposes a new approach for synthesising a predictive control for non-linear uncertain process based on a proposed reduced complexity discrete-time Volterra model known as GOBF-Volterra model. This model, provided by expanding each Volterra kernel on independent generalised orthonormal basis functions (GOBF), is efficient for the synthesis of non-linear model-based predictive control (NMBPC) which copes with physical constraints and geometrical constraints due to parameter uncertainties. A quadratic criterion is optimised and a new optimisation algorithm, formulated as a quadratic programming (QP) under linear and non-linear constraints, is proposed. Simulation results on a chemical reactor are presented to illustrate the performance of the proposed NMBPC strategy for uncertain process. This reveals that the stability performance of the resulting closed-loop system depends on the choice of the tuning parameters.

Optimal expansions of discrete-time bilinear models using Laguerre functions

In this paper, we propose a new reduced complexity model by expanding discrete-time bilinear model on Laguerre orthonormal bases. Thus the coefficients associated to the input, to the output and to the crossed product of the bilinear model are expanded on three independent Laguerre bases. The resulting model is entitled bilinear-Laguerre model with filters on model input and output. The parametric complexity reduction of the proposed model with respect to the classical bilinear model is proved theoretically. The structure and the parameter identification of the bilinear-Laguerre model is achieved by a new proposed approach which consists in solving an optimization problem built from the bilinear model without using system input/output observations. The performances of the proposed bilinear-Laguerre model and the proposed identification approach are illustrated on a numerical simulation and validated on a benchmark as the continuous stirred tank reactor system.

Nonlinear system modeling based on bilinear Laguerre orthonormal bases.

This paper proposes a new representation of discrete bilinear model by developing its coefficients associated to the input, to the output and to the crossed product on three independent Laguerre orthonormal bases. Compared to classical bilinear model, the resulting model entitled bilinear-Laguerre model ensures a significant parameter number reduction as well as simple recursive representation. However, such reduction still constrained by an optimal choice of Laguerre pole characterizing each basis. To do so, we develop a pole optimization algorithm which constitutes an extension of that proposed by Tanguy et al.. The bilinear-Laguerre model as well as the proposed pole optimization algorithm are illustrated and tested on a numerical simulations and validated on the Continuous Stirred Tank Reactor (CSTR) System.

Synthesis of a robust controller with reduced dimension by the Loop Shaping Design Procedure and decomposition based on Laguerre functions

In this paper, we present the synthesis of a robust controller for uncertain discrete systems. The synthesis method of such a robust controller is the generalization of the Loop Shaping Design Procedure (LSDP) approach of McFarlane and Glover in the discrete case based on the work of Gu et al. We exploit the bilinear transform known as Tustin’s method in order to formulate the discrete loop shaping technique. A discrete weighting filter and a shaped discrete plant result from this technique. By taking into account the coprime factor uncertainty representation for the resulting shaped plant and by applying the small gain theorem, we define the concept of the robust stabilization of the discrete LSDP approach. This concept is based on the resolution of an optimization problem characterized by the maximum stability margin for the synthesis of the robust controller. To calculate the robust controller we transform this problem to a standard robust H ∞ controller design based on the resolution of the Riccati equations. Also, we present the gap metric theory to characterize the controller’s robustness. We note that the resulting final controller is the combination of the discrete weighting filter and the robust controller. We propose then to exploit the recent work of Bouzrara et al. in order to develop a reduced robust controller by expanding the final controller on two independent Laguerre orthonormal bases. The discrete LSDP and the reduced controller approaches were validated on a Continuous Stirred Tank Reactor chemical reactor for a set of different equilibrium points in order to take into account the nonlinearities.

Online identification of the bilinear model expansion on Laguerre orthonormal bases

This paper proposes a new representation of discrete bilinear model by developing its coefficients associated to the input, to the output and to the crossed product on three independent Laguerre orthonormal bases. Compared to classical bilinear model, the resulting model entitled bilinear-Laguerre model ensures a significant parameter number reduction as well as simple recursive representation. However, this reduction is still subject to an optimal choice of the Laguerre poles defining the three Laguerre bases. Therefore, we propose an analytical solution to optimise the Laguerre poles which depend on Fourier coefficients defining the bilinear-Laguerre model, and that are identified using the regularised square error. The identification procedures of the Laguerre poles and Fourier coefficients are combined and carried out on a sliding window to provide an online identification algorithm of the bilinear-Laguerre model. The bilinear-Laguerre model as well as the proposed algorithm are illustrated and tested on a numerical simulation and validated on the continuous stirred tank reactor system.

Expansion of MIMO ARX model on Laguerre orthonormal bases

In this paper, we propose a new dynamic linear MIMO system representation by using discrete-time MIMO AutoRegressive with eXogenous input (ARX) model. To provide a reduced complexity model, each polynomial function of the MIMO ARX model associated to the inputs and to the outputs is expanded on independent Laguerre orthonormal basis to develop a new black-box linear MIMO ARX-Laguerre model. This reduction is ensured once the poles characterising each Laguerre orthonormal basis are set to their optimal values. Simulation results show the effectiveness of the proposed modeling method.

Reduced complexity Volterra model of non-linear MISO system

In this paper, we propose a new dynamic non-linear MISO system model using discrete-time Volterra series. To provide a reduced complexity model, each Volterra kernel is expanded on independent generalised orthonormal bases (GOBs) associated to the inputs to develop a new black-box non-linear MISO-GOB-Volterra model. However, this reduction is ensured once the poles characterising each independent generalised orthonormal basis (GOB) are set to their optimal values. For the selection of optimal GOBs poles, we develop two new general approaches based on Gauss-Newton and exhaustive algorithms, the performances of which are illustrated and compared in simulation.

Transition and control of nonlinear systems by combining the loop shaping design procedure and the gap metric theory

In this paper, in order to synthesize a control law we propose a new approach that enables identification of the intermediate equilibrium points of a nonlinear system, knowing the first and the last ones. These points are those around which the nonlinear system is linearized and therefore yields local models (sub-models) that contribute to forming the multimodel describing the nonlinear system. This approach is based on the transition from a given point (source) to the next by varying a scheduling parameter (SP) defining the source point sub-model. The variation of this parameter is limited by the maximum value of the stability margin determined by the loop shaping design procedure approach (LSDP) applied to such a sub-model. Hence, the new equilibrium point is defined by the new obtained value of the SP for which the gap metric between this sub-model and the one corresponding to the new value of SP is larger than the given stability margin. The different robust controllers synthesized for the different equilibrium points will be used to synthesize the robust control of the nonlinear system, by applying the gain-scheduling technique. The proposed transition approach as well as the robust control algorithm were validated on the continuous stirred tank reactor (CSTR) system.